Real Chern-Simons wave function

We examine the status of the Chern-Simons (or Kodama) state from the point of view of a formulation of gravity that uses only real connection and metric variables and a real action. We may package the real connection variables into the complex self-dual Ashtekar connection (and will do so to make contact with previous work), but that operation is essentially cosmetic and can be undone at any step or even bypassed altogether. The action will remain the (real) Einstein-Cartan action, forgoing the addition of the usual Hoist (or Nieh-Yan) term with an imaginary coefficient. It is then found that the constraints are solved by a modification of the Chern-Simons state which is a pure phase (in the Lorentzian theory, we stress), the phase containing only the fully gauge-invariant imaginary part of the Chern-Simons functional. Thus, the state for the real theory is nonpathological with regards to the most egregious criticisms facing its nonreal cousin, solving the complex theory. A straightforward modification of the real Chem-Simons state is also a solution in quasitopological theories based on the Euler invariant, for which the cosmological constant, Lambda, is dynamical. In that case it is enough to shift the usual factor of Lambda in the wave function to the inside of the spatial Chem-Simons integral. The trick only works for the quasi-Euler theory with a critical coupling previously identified in the literature. It does not apply to the quasi-Pontryagin theory.

Automated summary

This study examines the status of the Chern-Simons state from the perspective of gravity formulation using only real connection and metric variables with a real action. It is found that the constraints are solved by a modification of the Chern-Simons state, which is a pure phase. A straightforward modification of the real Chem-Simons state is also a solution in quasitopological theories based on the Euler invariant.